Heart sounds are critical for early detection of cardiovascular diseases, yet existing studies mostly focus on traditional signal segmentation, feature extraction, and shallow classifiers, which often fail to sufficiently capture the dynamic and nonlinear characteristics of heart sounds, limit recognition of complex heart sound patterns, and are sensitive to data imbalance, resulting in poor classification performance. To address these limitations, this study proposes a novel heart sound classification method that integrates improved Mel-frequency cepstral coefficients (MFCC) for feature extraction with a convolutional neural network (CNN) and a deep Transformer model. In the preprocessing stage, a Butterworth filter is applied for denoising, and continuous heart sound signals are directly processed without segmenting the cardiac cycles, allowing the improved MFCC features to better capture dynamic characteristics. These features are then fed into a CNN for feature learning, followed by global average pooling (GAP) to reduce model complexity and mitigate overfitting. Lastly, a deep Transformer module is employed to further extract and fuse features, completing the heart sound classification. To handle data imbalance, the model uses focal loss as the objective function. Experiments on two public datasets demonstrate that the proposed method performs effectively in both binary and multi-class classification tasks. This approach enables efficient classification of continuous heart sound signals, provides a reference methodology for future heart sound research for disease classification, and supports the development of wearable devices and home monitoring systems.